August 2008 Qualifying Examination Topology Problems 1-4 consist of true or false statements. Each statement is to be proved or disproved with brief but complete reasoning. Provide definitions of all underlined, italicized words and phrases. On page 2 find definitions and notations of some items appearing in the problems. There are 5 problems in total. Spacing: begin each problem on a new page. 1a) Any infinite set with the finite complement topologyis connected. b)The subspace of , where is connected. c)The real line with the lower limit topologyis connected. d)The complement of the “equatorial ” in the Mobius band is connected. 2 a) The set of integers with the topology generated by the basisis compact. b) The space defined in 2 a) is limit point compact. c) The set of all real invertible matrices is compact. d) The set of all real orthogonal matrices is compact. 3 a) Define a function from onto a three point set by With the resulting quotient topology induced by is Hausdorff. b) For any map

This
preview
has intentionally blurred sections.
Sign up to view the full version.